### Video Transcript

In this video, weβll learn how to
find missing lengths in a triangle containing two or three parallel lines using
proportionality. Weβll be able to recognize when
parallel lines inside or outside triangles create similar triangles, sometimes
called the side splitter theorem, use proportionality to find unknown lengths in
triangles with parallel lines, and weβll also look at the inverse of the side
splitter theorem.

Letβs start by recalling some
properties of parallel lines. We know, for example, that when two
parallel lines are cut by a transversal, the resulting corresponding angles are
equal. We know also that by adding a
second transversal, we can form two triangles. Now, giving each vertex a label, we
can define the larger triangle π΄π·πΈ and the smaller triangle π΄π΅πΆ.

Now, since the two pairs of
corresponding angles are equal, we can say that triangle π΄π·πΈ is similar to
triangle π΄π΅πΆ. And since these triangles are
similar, the ratios of their corresponding side lengths must be equal. This means π΄π΅ over π΄π·, thatβs
the ratio of π΄π΅ to π΄π·, is equal to π΄πΆ over π΄πΈ, which in turn is equal to
π΅πΆ over π·πΈ. So π΄π΅ is to π΄π· as π΄πΆ is to
π΄πΈ, and π΄πΆ is to π΄πΈ as π΅πΆ is to π·πΈ.

Weβre going to use these properties
of similar triangles in our first example to identify which pair of side lengths
have equal proportions when a triangle is cut by a line parallel to one of its
sides.

Using the diagram, which of the
following is equal to π΄π΅ over π΄π·. Is it (A) π΄πΆ over πΈπΆ or (B)
π΄π΅ over π·π΅ or (C) π΄π· over π·π΅, answer (D), π΄πΆ over π΄πΈ, or (E) π΄πΈ
over πΈπΆ.

We see from the diagram that
the base of triangle π΄πΈπ·, thatβ²s side πΈπ·, is parallel to the base of
triangle π΄π΅πΆ, thatβ²s side πΆπ΅. Since corresponding angles must
be equal if the two lines are parallel, angle π·πΈπ΄ is equal to angle π΅πΆπ΄,
and angle πΈπ·π΄ is equal to angle πΆπ΅π΄. Then side πΈπ· creates triangle
π΄π·πΈ, which is similar to the larger triangle π΄π΅πΆ. Since these triangles are
similar, the ratios of their corresponding side lengths must be equal. In particular, π΄πΈ over π΄πΆ
is equal to π΄π· over π΄π΅.

Now we want to find which of
the given fractions is equal to π΄π΅ over π΄π·. And we can do this by finding
the reciprocal of both sides of our equation, which gives us π΄πΆ over π΄πΈ is
equal to π΄π΅ over π΄π·. Hence, π΄π΅ over π΄π· is equal
to π΄πΆ over π΄πΈ, which corresponds to the given option (D).

In our next example, we see how to
find an unknown length in a triangle using proportions.

Find the value of π₯.

We see from the figure that the
lines π΄πΆ and π΄π΅ are transversals that intersect parallel lines π·πΈ and
π΅πΆ. And we know that the two pairs
of corresponding angles created by this intersection are equal. Thatβs angles π·πΈπ΄ and π΅πΆπ΄
and angles πΈπ·π΄ and πΆπ΅π΄. This being the case, we can say
the triangles π΄π΅πΆ and π΄π·πΈ are similar triangles since they each have the
common angle π΅π΄πΆ and their other two angles are also equal.

Now recall that when two
triangles are similar, the ratios of the length of their corresponding sides are
equal. In particular, π΄π· is to π΄π΅
as π·πΈ is to π΅πΆ. In other words, π΄π· over π΄π΅
is equal to π·πΈ over π΅πΆ. Now, we know that π΄π· is equal
to 10 units. π΄π΅ is equal to 10 plus 11
units, thatβs π΄π· plus π·π΅. π·πΈ is equal to 10 units, and
π΅πΆ is π₯. And so we have 10 over 10 plus
11 is equal to 10 over π₯. That is, 10 over 21 is equal to
10 over π₯.

Now, solving for π₯, we
multiply through by 21π₯ and divide both sides by 10, and we have π₯ equal to
21. Hence, using the given diagram,
we find that π₯ is equal to 21 units.

In the two previous examples, we
saw that if a line intersecting two sides of a triangle is parallel to the third
side, then the smaller triangle created by this line is similar to the original
triangle. And since triangles π΄π΅πΆ and
π΄π·πΈ are similar, we have that the proportions π΄π΅ over π΄π· and π΄πΆ over π΄πΈ
are equal. From the diagram, we can see also
that the line segment π΄π· can be split into π΄π΅ plus π΅π· and the line segment
π΄πΈ can be split into π΄πΆ plus πΆπΈ.

Now substituting these expressions
into our equation, we have π΄π΅ over π΄π΅ plus π΅π· is equal to π΄πΆ over π΄πΆ plus
πΆπΈ. And we can rearrange this to give
π΄π΅ times π΄πΆ plus πΆπΈ is equal to π΄πΆ times π΄π΅ plus π΅π·. If we next distribute our
parentheses and subtract π΄π΅ times π΄πΆ from both sides, we have π΄π΅ times πΆπΈ is
equal to π΄πΆ times π΅π·. And we can rearrange this to get
the equal proportions π΄π΅ over π΅π· is equal to π΄πΆ over πΆπΈ.

This leads us to the side splitter
theorem linking the line segments created when a parallel side is added to a
triangle. The side splitter theorem says that
if a line parallel to one side of a triangle intersects the other two sides of the
triangle, then the line divides those sides proportionally.

In our diagram, the side π΅πΆ is
parallel to the side π·πΈ. So we see that in our case the side
splitter theorem tells us that π΄π΅ is to π΅π· as π΄πΆ is to πΆπΈ. Note that this can be extended to
include parallel lines outside the triangle. We can form a similar triangle
outside the first triangle with a parallel line as shown in the diagram. And we can deduce an analog of the
side splitter theorem directly from these similar triangles.

In our next example, weβll see how
to use the side splitter theorem to identify proportional segments of the sides of
triangles so we can calculate a missing length.

In the figure, the sides ππ
and π΅πΆ are parallel. If π΄π equals 18, ππ΅ equals
24, and π΄π equals 27, what is the length of ππΆ?

Weβre given the side lengths
π΄π, ππ΅, and π΄π. And we want to find the length
of ππΆ. Weβre also told that the sides
ππ and π΅πΆ are parallel. Now, the side splitter theorem
tells us that if a line parallel to one side of a triangle intersects the other
two sides, then that line divides those two sides proportionally. In particular, in our case this
means that π΄π is to ππΆ as π΄π is to ππ΅. Now, if we substitute our known
lengths into this equation, π΄π is 27, π΄π is 18, and ππ΅ is 24, we have 27
over ππΆ is equal to 18 over 24. We can rearrange this to get
ππΆ equal to 24 over 18 times 27, which evaluates to 36.

And so using the side splitter
theorem, we find that the length of ππΆ is 36 units.

In our next example, weβll use the
side splitter theorem to help us solve a multistep problem involving triangles and
parallel lines.

The given figure shows a
triangle π΄π΅πΆ. (1) work out the value of
π₯. And (2) work out the value of
π¦.

Weβre given a triangle with a
line parallel to one side inscribed within it and various lengths of segments of
the sides of the triangle. And weβre asked to find the
value of π₯ and the value of π¦.

Letβs begin with part (1),
which is finding π₯, where we see the two of our side segments involve π₯. We note first that a line of
length two units inside the triangle is parallel to the side π΅πΆ. Now, the side splitter theorem
tells us that this line divides the two sides π΄πΆ and π΄π΅ proportionally.

Remember, according to the side
splitter theorem, if a line parallel to one side of a triangle intersects the
other two sides of the triangle, then the line divides those sides
proportionally. If we label this line segment
π·πΈ in our diagram, we could say that π΄π· over π·π΅ is equal to π΄πΈ over
πΈπΆ. Substituting in the given
lengths, we can form an equation which we can solve for π₯. Thatβs three over two π₯ plus
three is equal to two over π₯ plus five. Now, multiplying both sides by
π₯ plus five and two π₯ plus three and distributing the parentheses, we have
three π₯ plus 15 equals four π₯ plus six. Subtracting three π₯ and six
from both sides and swapping sides, we then have π₯ is equal to nine.

So now, making a note of this
and making some space, we can use this value of π₯ in part (2) of the question
to find the value of π¦. Now, since the two triangles
created by the intersection of side π·πΈ share a common angle π΄ and the pairs
of corresponding angles also created by this line are equal, we can say the
triangles π΄π·πΈ and π΄π΅πΆ are similar triangles. In particular, this means that
the proportions π΄π· over π΄π΅ and π·πΈ over π΅πΆ are equal. Now, we know that π΄π΅ is equal
to the sum π΄π· plus π·π΅. And thatβs three plus two π₯
plus three. And we have π₯ equal to nine
from part (1). And so this evaluates to
24.

So now, substituting our values
π΄π· equals three, π΄π΅ equals 24, π·πΈ equals two, and π΅πΆ equals π¦ into our
equation, we have three over 24 is equal to two over π¦. Now, multiplying both sides by
π¦ and by 24 over three, we have π¦ equal to 16. Hence, from the given figure,
we find that π₯ is equal to nine and π¦ is equal to 16.

So far, weβve used the side
splitter theorem to solve problems of proportionality and triangles. Weβve also learned that this can be
extended to parallel lines outside a given triangle. In fact, it turns out that the
converse of this result is also true. And this can be very useful when
solving problems of this type. The converse of the side splitter
theorem tells us that if a line intersecting two sides of a triangle splits these
sides into equal proportions, then that line must be parallel to the third side of
the triangle.

In all three of the diagrams shown,
π΄π΅πΆ is a triangle and the line π·πΈ intersects π΄π΅ at π· and π΄πΆ at πΈ. In other words, if proportions π΄π·
over π·π΅ and π΄πΈ over πΈπΆ are equal, then the lines π·πΈ and π΅πΆ must be
parallel. So letβs now use the converse of
the side splitter theorem to find unknown lengths in a given triangle.

Given that π΄π΅πΆπ· is a
parallelogram, find the length of ππ.

To find the length of the side
ππ of the smaller triangle πππ inside the parallelogram, letβs begin by
identifying some relevant information about the two triangles πππ and
ππ·πΆ. Weβre given that ππ is equal
to ππ· and ππ is equal to ππΆ. We also recall that the side
splitter theorem tells us that if a line parallel to one side of a triangle
intersects the other two sides, then the line divides those sides
proportionally. Conversely if a line splits two
sides of a triangle into equal proportions, then that line must be parallel to
the third side.

In our case, since sides ππ·
and ππΆ of the larger triangle ππ·πΆ have been divided into equal proportions,
we can apply the converse of the side splitter theorem to deduce that sides π·πΆ
and ππ must be parallel. Now, recall also that if a line
parallel to a side of a triangle intersects the other two sides, as does the
line πΌπ½ in the diagram, then the smaller triangle created by that line, here
triangle πΌπ½π», is similar to the original triangle πΉπΊπ». In our case, this means the
triangle πππ is similar to triangle ππ·πΆ.

Now, since the sides π·πΆ and
π΄π΅ are opposite sides of the parallelogram π΄π΅πΆπ·, they must have equal
length. So now, we know that side π·πΆ
has length 134.9 centimeters. Now, making some space, if we
extract our triangle and call the length ππ lowercase π₯, our side length ππ·
is equal to two π₯. Since our triangles πππ and
ππ·πΆ are similar, we can form an equation with the side lengths are shown such
that the proportions ππ over ππ· and ππ over π·πΆ are equal.

Now, substituting in the side
lengths we know, we have π₯ over two π₯ is equal to ππ over 134.9. We can divide top and bottom on
the left by π₯ to give one over two is equal to ππ over 134.9. Then multiplying through by
134.9, we have ππ equals 134.9 over two, which evaluates to 67.45. The length ππ is therefore
67.45 centimeters.

Letβs now, finish by recapping some
of the main points weβve covered. If a line intersecting two sides of
a triangle is parallel to the remaining side, then the smaller triangle created by
this line is similar to the original triangle. The side splitter theorem tells us
that if a line parallel to one side of a triangle intersects the other two sides of
the triangle, then the line divides those sides proportionally.

The side splitter theorem extends
to include parallel lines outside a triangle. If a line outside a triangle is
parallel to one of the sides of the triangle and intersects the extensions of the
other two sides, then the line divides the extensions of those sides
proportionally. And finally, the converse of the
side splitter theorem states that if a line splits two sides of a triangle
proportionally, then that line is parallel to the remaining side.